N = Matrix or Vector

Weighting on the state-input multiplication term in the cost function. If omitted, a zero matrix with appropriate dimensions will be considered.

poles = true or false

True means the eigenvalues of A-BK are returned. The default value is false.    

riccati = true or false

True means the solution of the associated Riccati equation is returned.

return_Kr = true or false

True means the direct gain Kr is returned. The default value is false.

parameters = {list, set}(name = complexcons)

Specifies numeric values for the parameters of sys. These values override any parameters previously specified for sys. The numeric value on the right-hand side of each equation is substituted for the name on the left-hand side in the sys equations. The default is the value of sys given by DynamicSystems:-SystemOptions(parameters).

The pair $\left(A,B\right)$ must be stabilizable.

The pair $\left(Q-N{R}^{-1}{N}^T,A-B{R}^{-1}{N}^T\right)$ must have no unobservable modes on the imaginary axis in continuous-time domain or on the unit circle in discrete-time domain.

$R>0$ (positive definite) and $Q-N{R}^{-1}{N}^T\ge 0$ (positive semidefinite).

The LQR command calculates the LQR state feedback gain for a system.

The system sys is a continuous or discrete time linear system object created using the DynamicSystems package. The system object must be in state-space (SS) form. The state-space system can be either single-input/single-output (SISO) or multiple-input/multiple-output (MIMO).

In continuous time, the optimal state feedback gain, $K$, is calculated such that the quadratic cost function

In discrete time, the optimal state feedback gain, $K$, is calculated such that the quadratic cost function

Q and R are expected to be symmetric. If the input Q and/or R are not symmetric, their symmetric part will be considered since their antisymmetric (skew-symmetric) part has no role in the quadratic cost function.

In addition to the state feedback gain, depending on the corresponding option values, the command also returns the closed-loop eigenvalues and the solution of the associated Riccati equation.

The direct gain $\mathrm{Kr}$ is computed as follows:

If sys contains structured uncontrollable states, they are removed before computing the LQR state feedback. The resulting gain $K$ is then filled with zeros at positions corresponding to the removed states; however, the other outputs are not filled and, consequently, they may have lower dimensions as expected.

$\mathrm{with}\left(\mathrm{ControlDesign}\right)\:$

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{csys}≔\mathrm{NewSystem}\left(\mathrm{Matrix}\left(\left[\left[\frac{2}{s^2+3s+1}\,\frac{1}{s+2}\right]\,\left[\frac{s-1}{s^2+5}\,\frac{7}{\left(s+1\right)\left(s+4\right)}\right]\right]\right)\right)\:$

$\mathrm{csys}:-\mathrm{tf}$

$\left[\begin{array}{cc}\frac{2}{s^2+3s+1}& \frac{1}{s+2}\\ \frac{s-1}{s^2+5}& \frac{7}{s^2+5s+4}\end{array}\right]$

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{csys}\right)\:$

$\mathrm{sys}:-a\;$$\mathrm{sys}:-b\;$$\mathrm{sys}:-c\;$$\mathrm{sys}:-d$

$\left[\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ \mathrm{-5}& \mathrm{-15}& \mathrm{-6}& \mathrm{-3}& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& \mathrm{-8}& \mathrm{-14}& \mathrm{-7}\end{array}\right]$

$\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 1& 0\\ 0& 0\\ 0& 0\\ 0& 1\end{array}\right]$

$\left[\begin{array}{ccccccc}10& 0& 2& 0& 4& 5& 1\\ \mathrm{-1}& \mathrm{-2}& 2& 1& 14& 7& 0\end{array}\right]$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$Q≔\frac{1}{3}\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(7\right)\;$$R≔2\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(2\right)$

$Q≔\left[\begin{array}{ccccccc}\frac{1}{3}& 0& 0& 0& 0& 0& 0\\ 0& \frac{1}{3}& 0& 0& 0& 0& 0\\ 0& 0& \frac{1}{3}& 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{3}& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{3}& 0& 0\\ 0& 0& 0& 0& 0& \frac{1}{3}& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{3}\end{array}\right]$

$R≔\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$K≔\mathrm{LQR}\left(\mathrm{sys}\,Q\,R\right)$

$K≔\left[\begin{array}{ccccccc}0.0166389810974705& 0.299443675955044& 0.867776452191478& 0.301850931076329& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.0104098937986105& 0.0195439507289429& 0.0146813589873455\end{array}\right]$

$\mathrm{Kpr}≔\mathrm{LQR}\left(\mathrm{sys}\,Q\,R\,'\mathrm{poles}'=\mathrm{true}\,'\mathrm{riccati}'=\mathrm{true}\,'\mathrm{return\_Kr}'=\mathrm{true}\right)\:$$\mathrm{Kpr}\left[1\right]\;$$\mathrm{Kpr}\left[2\right]\;$$\mathrm{Kpr}\left[3\right]\;$$\mathrm{Kpr}\left[4\right]$

$\left[\begin{array}{ccccccc}0.0166389810974705& 0.299443675955044& 0.867776452191478& 0.301850931076329& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.0104098937986105& 0.0195439507289429& 0.0146813589873455\end{array}\right]$

$\left[\begin{array}{c}\mathrm{-2.62756665299744}+0.\mathrm{I}\\ \mathrm{-0.145624709217511}+2.22784297750641\mathrm{I}\\ \mathrm{-0.145624709217511}-2.22784297750641\mathrm{I}\\ \mathrm{-0.383034859643869}+0.\mathrm{I}\\ \mathrm{-1.00185981437603}+0.\mathrm{I}\\ \mathrm{-1.98515367555330}+0.\mathrm{I}\\ \mathrm{-4.02766786905801}+0.\mathrm{I}\end{array}\right]$

$\left[\begin{array}{ccccccc}3.5035710678024963642735698334738086970173092871078& 8.9063101270541111531167581075671803104947792461458& 3.1283881812209808268605882029451459276062782436105& 0.033277962194940937420660323652668921352991539626009& 0.& 0.& 0.\\ 8.9063101270541111531167581075671803104947792461458& 27.017929837486759850611800937425078242698100086181& 10.999686730609094484906069375902437265430236365035& 0.59888735191008724334359278295378054778377357379270& 0.& 0.& 0.\\ 3.1283881812209808268605882029451459276062782436105& 10.999686730609094484906069375902437265430236365035& 8.7538607942749524452688978919514869492958202135138& 1.7355529043829555120085717535997438572353549848779& 0.& 0.& 0.\\ 0.033277962194940937420660323652668921352991539626009& 0.59888735191008724334359278295378054778377357379270& 1.7355529043829555120085717535997438572353549848779& 0.60370186215265844164465469332087994051094168275939& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.60458713892716850967689842079379231322521721813087& 0.38094591975383059966396461879896053218176570663419& 0.020819787597221039613645137137870970189590359262787\\ 0.& 0.& 0.& 0.& 0.38094591975383059966396461879896053218176570663419& 0.66444743776701954149815236150708781014928590795157& 0.039087901457885851258326665942712298638987275213164\\ 0.& 0.& 0.& 0.& 0.020819787597221039613645137137870970189590359262787& 0.039087901457885851258326665942712298638987275213164& 0.029362717974691029572229097828269653238134822396934\end{array}\right]$

$\left[\begin{array}{cc}0.487728789828921& \mathrm{-0.139351082808263}\\ 0.0556278464847126& 0.556278464847126\end{array}\right]$

$\mathrm{dsys}≔\mathrm{ToDiscrete}\left(\mathrm{csys}\,1\,'\mathrm{method}'=\mathrm{bilinear}\right)$

$\mathrm{dsys}≔\left[\begin{array}{c}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; 1}\\ \mathrm{2\; output(s);\; 2\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\,\mathrm{u2}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\,\mathrm{y2}\left(z\right)\right]\end{array}\right.$

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{dsys}\right)$

$\mathrm{sys}≔\left[\begin{array}{c}\mathbf{State\; Space}\\ \mathrm{discrete;\; sampletime\; =\; 1}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 7\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(q\right)\,\mathrm{u2}\left(q\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(q\right)\,\mathrm{y2}\left(q\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(q\right)\,\mathrm{x2}\left(q\right)\,\mathrm{x3}\left(q\right)\,\mathrm{x4}\left(q\right)\,\mathrm{x5}\left(q\right)\,\mathrm{x6}\left(q\right)\,\mathrm{x7}\left(q\right)\right]\end{array}\right.$

$K≔\mathrm{LQR}\left(\mathrm{sys}\,Q\,R\right)$

$K≔\left[\begin{array}{ccccccc}0.0481202313599143& 0.301603484287272& \mathrm{-0.420834895345035}& 0.0511514301656090& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.& 0.0372408140701274& \mathrm{-9.76066563662570}\times {10}^{\mathrm{-51}}\end{array}\right]$

$\mathrm{Kpr}≔\mathrm{LQR}\left(\mathrm{sys}\,Q\,R\,'\mathrm{poles}'=\mathrm{true}\,'\mathrm{riccati}'=\mathrm{true}\,'\mathrm{return\_Kr}'=\mathrm{true}\right)\:$$\mathrm{Kpr}\left[1\right]\;$$\mathrm{Kpr}\left[2\right]\;$$\mathrm{Kpr}\left[3\right]\;$$\mathrm{Kpr}\left[4\right]$

$\left[\begin{array}{ccccccc}0.0481202313599143& 0.301603484287272& \mathrm{-0.420834895345035}& 0.0511514301656090& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.& 0.0372408140701274& \mathrm{-9.76066563662570}\times {10}^{\mathrm{-51}}\end{array}\right]$

$\left[\begin{array}{c}\mathrm{-0.0959924471365448}+0.725780367567049\mathrm{I}\\ \mathrm{-0.0959924471365448}-0.725780367567049\mathrm{I}\\ 0.597646681582207+0.\mathrm{I}\\ \mathrm{-0.133580894274726}+0.\mathrm{I}\\ 0.+0.\mathrm{I}\\ \mathrm{-0.271790906819696}+0.\mathrm{I}\\ 0.271790906819696+0.\mathrm{I}\end{array}\right]$

$\left[\begin{array}{ccccccc}0.34208246630795069777141693271855815627313838388516& 0.054836997143688792408247958303061879045638516173491& \mathrm{-0.076515435518044319517218299335460057103299438798401}& 0.0093002600302037228693027858413961837865887986834760& 0.& 0.& 0.\\ 0.054836997143688792408247958303061879045638516173491& 1.0190795442103837012531041373825245474117656429532& \mathrm{-0.42467264975475382590709073532181913929136909457623}& \mathrm{-0.016116727827330943226778991445093684120949837718847}& 0.& 0.& 0.\\ \mathrm{-0.076515435518044319517218299335460057103299438798401}& \mathrm{-0.42467264975475382590709073532181913929136909457623}& 2.0214621985616492009289490977082087182086468171616& \mathrm{-0.50965995705126921942600611836937653744093332547732}& 0.& 0.& 0.\\ 0.0093002600302037228693027858413961837865887986834760& \mathrm{-0.016116727827330943226778991445093684120949837718847}& \mathrm{-0.50965995705126921942600611836937653744093332547732}& 2.2491943868514685137656620258975394150765552695306& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.33333333333333333333333333333333333333333333333333& \mathrm{-3.8496822040751973138321389112729645857823238931349}\times {10}^{\mathrm{-50}}& \mathrm{-6.1340857535650804575141297226367399037182210044452}\times {10}^{\mathrm{-51}}\\ 0.& 0.& 0.& 0.& \mathrm{-3.8496822040751973138321389112729645857823238931349}\times {10}^{\mathrm{-50}}& 0.67494240312586739704767351946529754472265510527072& \mathrm{-2.9362773606352192932407413788839969812745930630482}\times {10}^{\mathrm{-50}}\\ 0.& 0.& 0.& 0.& \mathrm{-6.1340857535650804575141297226367399037182210044452}\times {10}^{\mathrm{-51}}& \mathrm{-2.9362773606352192932407413788839969812745930630482}\times {10}^{\mathrm{-50}}& 1.0082757364592007303810068527986308780559884386041\end{array}\right]$

$\left[\begin{array}{cc}0.474104074310421& \mathrm{-0.135458306945628}\\ 0.0578831064518730& 0.578831064350061\end{array}\right]$